Long Wave Interaction with a Partially Immersed Body. Part I: Mathematical Models

In the present article we consider the problem of wave interaction with a partially immersed, but floating body. We assume that the motion of the body is prescribed. The general mathematical formulation for this problem is presented in the framework of a hierarchy of mathematical models. Namely, in this first part we formulate the problem at every hierarchical level. The special attention is payed to fully nonlinear and weakly dispersive models since they are most likely to be used in practice. For this model we have to consider separately the inner (under the body) and outer domains. Various approached to the gluing of solutions at the boundary is discussed as well. We propose several strategies which ensure the global conservation or continuity of some important physical quantities.

[1]  J. Lee,et al.  A VISCOUS ROTATIONAL MODEL FOR WAVE OVERTOPPING OVER MARINE STRUCTURE , 1997 .

[2]  D. Dutykh,et al.  Peregrine’s System Revisited , 2018, 1802.03301.

[3]  David M. Skene,et al.  Reflection and transmission of regular water waves by a thin, floating plate , 2017 .

[4]  F. Serre,et al.  CONTRIBUTION À L'ÉTUDE DES ÉCOULEMENTS PERMANENTS ET VARIABLES DANS LES CANAUX , 1953 .

[5]  Pengzhi Lin,et al.  A numerical study of solitary wave interaction with rectangular obstacles , 2004 .

[6]  Leon Bieber,et al.  Numerical Schemes For Conservation Laws , 2016 .

[7]  James Serrin,et al.  Mathematical Principles of Classical Fluid Mechanics , 1959 .

[8]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[9]  H. Holden,et al.  Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs , 2010 .

[10]  Hideaki Miyata,et al.  Finite-difference simulation of nonlinear ship waves , 1985 .

[11]  Pengzhi Lin,et al.  A multiple-layer σ-coordinate model for simulation of wave–structure interaction , 2006 .

[12]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[13]  H. J.,et al.  Hydrodynamics , 1924, Nature.

[14]  T. Katsaounis,et al.  Dispersive wave runup on non-uniform shores , 2011, 1101.1729.

[15]  O. Phillips.  The equilibrium range in the spectrum of wind-generated waves , 1958, Journal of Fluid Mechanics.

[16]  F. Ursell,et al.  The long-wave paradox in the theory of gravity waves , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  W. Rankine.  On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance. [Abstract] , 1869 .

[18]  R. Fetecau,et al.  On a regularization of the compressible Euler equations for an isothermal gas , 2009 .

[19]  D. Dutykh,et al.  PRACTICAL USE OF VARIATIONAL PRINCIPLES FOR MODELING WATER WAVES , 2010, 1002.3019.

[20]  Volker Roeber,et al.  Boussinesq-type model for nearshore wave processes in fringing reef environment , 2010 .

[21]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[22]  Energy equation for certain approximate models of long-wave hydrodynamics , 2014 .

[23]  M. Gharib,et al.  Resonance wave pumping with surface waves , 2015, Journal of Fluid Mechanics.

[24]  P. Milewski,et al.  Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations , 2011, European Journal of Applied Mathematics.

[25]  Ming Zhao,et al.  Numerical simulation of solitary wave scattering by a circular cylinder array , 2007 .

[26]  Michael Isaacson.  Nonlinear-wave effects on fixed and floating bodies , 1982 .

[27]  Ljf Lambert Broer.  On the hamiltonian theory of surface waves , 1974 .

[28]  M. Boussinesq.  Essai sur la théorie des eaux courantes , 1873 .

[29]  Denys Dutykh,et al.  The VOLNA code for the numerical modeling of tsunami waves: Generation, propagation and inundation , 2010, 1002.4553.

[30]  D. Dutykh,et al.  Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space , 2017, 1706.08815.

[31]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[32]  Claude Marmignon,et al.  Choice of measure source terms in interface coupling for a model problem in gas dynamics , 2016, Math. Comput..

[33]  Finn Løvholt,et al.  Oceanic propagation of a potential tsunami from the La Palma Island , 2008 .

[34]  Nektarios Chrysoulakis,et al.  Mathematical Modelling of Tsunami Waves , 2010, ERCIM News.

[35]  Eleuterio F. Toro,et al.  Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry , 2008, J. Comput. Phys..

[36]  L. Gosse,et al.  A Numerical Glimpse at Some Non-standard Solutions to Compressible Euler Equations , 2017 .

[37]  G. M. Tuynman,et al.  Structure of Dynamical Systems A Symplectic View of Physics , 1997 .

[38]  Variational principle for approximate models of wave hydrodynamics , 1996 .

[39]  Claes Eskilsson,et al.  A stabilised nodal spectral element method for fully nonlinear water waves , 2015, J. Comput. Phys..

[40]  Dimitrios Mitsotakis,et al.  Theory and Numerical Analysis of Boussinesq Systems: A Review , 2008 .

[41]  Sijue Wu,et al.  Global wellposedness of the 3-D full water wave problem , 2011 .

[42]  A Leray-type regularization for the isentropic Euler equations , 2007 .

[43]  M Kashiwagi.  Non-linear simulations of wave-induced motions of a floating body by means of the mixed Eulerian-Lagrangian method , 2000 .

[44]  Alfio Quarteroni,et al.  Mathematical models and numerical simulations for the America"s Cup , 2005 .

[45]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[46]  M. Longuet-Higgins,et al.  Accelerations in Steep Gravity Waves , 1985 .

[47]  L. Shemer,et al.  On Kinematics and Dynamics of Breaking Water Waves , 2013 .

[48]  Hierarchy of nonlinear models of the hydrodynamics of long surface waves , 2015 .

[49]  A. J. Hermans.  Theory of Water Waves , 2011 .

[50]  Denys Dutykh,et al.  Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler equations) , 2017, Commun. Nonlinear Sci. Numer. Simul..

[51]  C. Linton,et al.  Handbook of Mathematical Techniques for Wave/Structure Interactions , 2001 .

[52]  Patrick Bar-Avi,et al.  Water Waves: The Mathematical Theory with Application , 1995 .

[53]  An overview of some recent results on the Euler system of isentropic gas dynamics , 2015, 1508.02937.

[54]  Denys Dutykh,et al.  Numerical modelling of surface water wave interaction with a moving wall , 2017, ArXiv.

[56]  Pavel Tkalich,et al.  Tsunami propagation modelling – a sensitivity study , 2007 .

[57]  D. Dutykh,et al.  Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry , 2017, 1707.01304.

[58]  合田 良実,et al.  Random seas and design of maritime structures , 1985 .

[59]  Scott Draper,et al.  Identifying the design wave group for the extreme response of a point absorber wave energy converter , 2016 .

[60]  Denys Dutykh,et al.  Energy of tsunami waves generated by bottom motion , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[61]  Denys Dutykh,et al.  Numerical solution of conservation laws on moving grids , 2015, ArXiv.

[62]  Denys Dutykh,et al.  Finite volume schemes for dispersive wave propagation and runup , 2010, J. Comput. Phys..

[63]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

[64]  Ondrej Kreml,et al.  Global Ill‐Posedness of the Isentropic System of Gas Dynamics , 2013, 1304.0123.

[65]  Tao Pang,et al.  An Introduction to Computational Physics , 1997 .

[66]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[67]  P. M. Naghdi,et al.  A derivation of equations for wave propagation in water of variable depth , 1976, Journal of Fluid Mechanics.

[68]  Som Deo Sharma,et al.  Influence of Canal Topography on Ship Waves in Shallow Water , 2001 .

[69]  Peter A. Troch,et al.  A Review of Numerical Modelling of Wave Energy Converter Arrays , 2012 .

[70]  Maurizio Brocchini,et al.  A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[71]  Fayssal Benkhaldoun,et al.  Exact solutions to the Riemann problem of the shallow water equations with a bottom step , 2001 .

[72]  Thierry Colin,et al.  Long Wave Approximations for Water Waves , 2005 .

[73]  Denys Dutykh,et al.  Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling , 2015, Geosciences.

[74]  Yi Li.  A shallow‐water approximation to the full water wave problem , 2006 .

[75]  G. Wu,et al.  Interactions of fully nonlinear solitary wave with a freely floating vertical cylinder , 2016 .

[76]  J. J. Stoker.  Water Waves: The Mathematical Theory with Applications , 1957 .

[77]  D. Dutykh,et al.  Dispersive shallow water wave modelling. Part IV: Numerical simulation on a globally spherical geometry , 2017, 1707.02552.

[78]  John W. Miles,et al.  Weakly dispersive nonlinear gravity waves , 1985, Journal of Fluid Mechanics.

[79]  Muhammad R. Hajj,et al.  Reflection and Transmission of Waves over Submerged Breakwaters , 2001 .

[80]  G. Pedersen,et al.  Coupling of Dispersive Tsunami Propagation and Shallow Water Coastal Response , 2010 .

[81]  Denys Dutykh,et al.  Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics , 2015, Commun. Nonlinear Sci. Numer. Simul..